A printable schedule can be downloaded in the pdf format from the web site of Ligne d'Azur, Nice's local bus company. However for convenience we are repeating this information here with additional details.
Bus 84 leaves the station in Beaulieu for Nice, passing the Observatory, every day except Sunday every 30 min from 7:00 to 20:00 (thus departures are at 7:00, 7:30, 8:00, …, 20:00).
Bus 84 leaves the Riquier station (east Nice) for Beaulieu, passing the Observatory, every day except Sunday at the hour+20 min and the hour+50 min. The last bus leaves Riquier at 18:20. Typically the bus takes 10-15 min to reach the Observatory.
Both terminal bus stations are in front of the respective railway stations:
Bus stops should be explicitly requested by pressing one of 3-4 red buttons placed inside the bus near windows and the entrance door.
Single-trip tickets are on sale from bus drivers for 1.30 Euro (drivers strongly prefer exact change).
Additional information (not directly relevant unless people want to go to Nice):
The bus from the Observatory to Riquier station is available every hour+15 min and 45 min, e.g., at 16:15, 16:45, 17:15 and so on till 19:45. Often the bus is 5 minutes (or a little more) late. it is possible to return to Beaulieu after 18:30 by way of bus to Riquier + train to Beaulieu.